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IMPORTANT: This assignment is assessed, and carries weight 10% towards
your final mark for this unit. The deadline for submission of work for this assignment
is 11:55pm, Monday the 27th of May 2024. You should submit your work to the
Moodle submission box.
It is stressed that in your submission, you must not use generative artificial
intelligence (AI) to generate any materials or content in relation to your answers.
If you do have questions about your submission, please contact Miss Ruofan Xu
at ruofan.xu@monash.edu.

1. Recall the OECD annual dataset for their health expenditures of 18 countries for the period of 1990–2012. Let Uit be the Gross Domestic Production
   (GDP) per capita (using constant 2005 PPP) for country i at year t, and
   Yit = log 
   Uit
   Ui,t−1
   
   be the growth of the logarithm of the GDP per capita.

• Consider the following dynamic panel data model:
  Yit = βi Yi,t−1 + αi + εit (1)
  for 3 ≤ t ≤ T and 1 ≤ i ≤ N, where each βi
  is an unknown parameter,
  αi
  is the fixed effects component, and {εit} is an array of independent
  and identically distributed (i.i.d.) random errors with E[εit] = 0 and
  E[ε
  2
  it] = σ
  2
  1 < ∞.
  As discussed in the Lectures, β and αi can be estimated1 by
  bβiIV =
  X
  T
  t=3
  (Yi,t−1 − Yi,t−2)Yi,t−2
  !−1 X
  T
  t=3
  (Yit − Yi,t−1)Yi,t−2
  !
  , (2)
  αbiIV = Yi· − bβiIV Yi,−1, (3)
  where Yi· =
  1
  T −2
  PT
  t=3 Yit and Yi,−1 =
  1
  T −2
  PT
  t=3 Yi,t−1.
  – Use equations (2) and (3) to estimate βi and αi
  . [7 marks]
  – Plot the estimates, bβiIV and αbiIV, versus 1 ≤ i ≤ 18 in one picture.
  [7 marks]
  – What would be your interpretations from the visualisation of the
  estimates according to the individual countries ? [6 marks]
  1
  If needed, you may use a slightly modified version of the code used in Question 3 of Tutorial
  9 in Week 9.
  1

1. Consider a trending dynamic panel data model of the form:
   Yit = β0 + β1 t + β2 t
   2 + γ Yi,t−1 + αi + εit (4)
   for 1 ≤ i ≤ N, 2 ≤ t ≤ T, where β = (β0, β1, β2)
   0
   is a vector of unknown parameters, αi
   is the fixed effects component identically distributed
   with E[αi
   ] = 0 and 0 < E[α
   2
   i
   ] < ∞, {εit : 1 ≤ i ≤ N; 2 ≤ t ≤ T} is an array
   of i.i.d. random errors with E[ε11] = 0 and 0 < E [ε
   2
   11] = σ
   2
   1 < ∞, and is
   independent of {(αi
   , Yi,t−1)}.

• Using one of the estimation methods you have learned from the lectures,
  write down detailed formulae for the estimators, denoted by bβj and γb, of
  the unknown parameters βj
  for j = 0, 1, 2 and γ, respectively. [10 marks]
• Discuss how to test H0 : γ = 0 and provide the main steps to show how
  to implement the suggested test of yours. [10 marks]
• Under H0, outline the main steps for the estimation of βj
  for j = 0, 1, 2.
  [10 marks]

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